Existence of Nontrivial Solutions for a Critical Perturbed Quasilinear Elliptic System

Huixing Zhang,Xiangzhi Zhang

doi:10.1155/2014/758963

Huixing Zhang, Xiangzhi Zhang

Open Access

https://doi.org/10.1155/2014/758963

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Abstract

We consider a perturbed quasilinear elliptic system involving thep-Laplacian with critical growth terms inRN. Under proper conditions, we establish the existence of nontrivial solutions by using the variational methods.

Highlights

We are concerned with the following perturbed quasilinear elliptic system involving the p-Laplacian:
V, x ∈ Ω, where λ, μ > 0, α > 1, β > 1 satisfy α + β = p∗ and p∗ = Np/(N − p) denotes the critical Sobolev exponent. He proved that the problem (4) has at least two positive solutions in W01,p(Ω) × W01,p(Ω)
In [17], we mostly focus on discussing the properties of the functions Hu(u, V), HV(u, V) and the associated primitive function H(u, V) which bring some difficulties in proving that Iλ(u, V) satisfies the compactness condition

Summary

Introduction

We are concerned with the following perturbed quasilinear elliptic system involving the p-Laplacian:. Where 1 < q < 2, α > 1, β > 1 satisfy 2 < α + β < 2∗ and the functions f(x), g(x), h(x) satisfy some suitable conditions He established the existence of at least two positive solutions for the problem (3) when the pair of the parameters (λ, μ). Hsu and Lin [15] considered a similar problem and proved that the problem (3) has at least two positive solutions in W01,2(Ω) × W01,2(Ω) involving critical exponents. V, x ∈ Ω, where λ, μ > 0, α > 1, β > 1 satisfy α + β = p∗ and p∗ = Np/(N − p) denotes the critical Sobolev exponent He proved that the problem (4) has at least two positive solutions in W01,p(Ω) × W01,p(Ω).

Notations and Preliminaries

Proof of the Main Result